Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to X$ that has some fine number of double points. I am interested in knowing how to compute minimum number of such double points, minimized over all such immersed disks. For now, I will call it the immersion number of $K$ - denoted $I(K)$, for fun. Has this quantity been studied in the literature?
One thing to note about $I(K)$ is that it provides a lower bound on the minimum genus of an orientable surface bounding $K$ (the "4-ball genus" if $X = B^4$). I am interested in knowing examples of knots (even/especially with $X = B^4$) where $I(K)$ and the minimum genus of an orientable surface properly embedded in $X$ bounding $K$ are not equal.